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Computationally-Efficient Approximation Mechanisms

Computationally-Efficient Approximation Mechanisms. Computationally-Efficient Approximation Mechanisms. Algorithms in Computer Science, and Mechanisms in Game Theory, are remarkably similar objects . But the resulting two sets of properties are completely different .

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Computationally-Efficient Approximation Mechanisms

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  1. Computationally-Efficient Approximation Mechanisms

  2. Computationally-Efficient Approximation Mechanisms Algorithms in Computer Science, and Mechanisms in Game Theory, are remarkably similar objects. But the resulting two sets of properties are completely different. We would like to merge them – to simultaneously exhibit “good” game theoretic properties as well as “good” computational properties. Monotonicity and Implementability

  3. Computationally-Efficient Approximation Mechanisms • A social choice setting – reminder • Two monotonicity conditions • Cyclic monotonicity • representation graph of a social choice function • Weak monotonicity • Weak monotonicity in Order-based domain • Single-Dimensional Domains and Job Scheduling • Scheduling related machines • single-dimensional linear domains • Summary Outline:

  4. Computationally-Efficient Approximation Mechanisms A finite set . Each player has a type (valuation function) Goal: find dominant strategy: social choice function: Requirement: price function: s.t: Reminder: A social choice setting

  5. Computationally-Efficient Approximation Mechanisms • A social choice setting – reminder • Two monotonicity conditions • Cyclic monotonicity • representation graph of a social choice function • Weak monotonicity • Weak monotonicity in Order-based domain • Single-Dimensional Domains and Job Scheduling • Scheduling related machines • single-dimensional linear domains • Summary Outline:

  6. Computationally-Efficient Approximation Mechanisms • Cyclic monotonicity • Fix a playerand • Assume w.l.o.g is onto • Dominant strategy: • Prices in are now constants: Two monotonicity conditions

  7. Computationally-Efficient Approximation Mechanisms • Cyclic monotonicity • Need to find s.t Definition: Two monotonicity conditions Motivation: If we’ll show that then

  8. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity This can easily solved by looking at the representation graph Definition:Representation graph The representation graph of a social choice function is a directed weighted graph where and . The weight of an edge(for ) is

  9. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Representation graph example: Single Player Lets build the representation graph:

  10. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Representation graph example: and . Calculating: -1

  11. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Representation graph example: and . Calculating: 2 -1

  12. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Proposition: There exists a feasible assignment to the representation graph has no negative-length cycles. Assignment: set to the length of the shortest path from to some arbitrary fixed node .

  13. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Proof: no negative-length cycles. Suppose is a negative cycle, i.e. and

  14. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Proof cont: no negative-length cycles. Suppose every cycle is non-negative. Fix arbitrary and set = length of the shortest path from to (well defined). The shortest path from to (= ) is no longer than + the shortest path from to (= ). i.e.

  15. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Definition:Cyclic monotonicity A social choice function satisfies cyclic monotonicity if for every player , some integerand Where forand Proposition: satisfies Cyclic monotonicity the representation graph of has no negative cycles

  16. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Proof: • satisfies cyclic monotonicity definition

  17. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Proof cont: • satisfies cyclic monotonicity • Suppose is a negative cycle, i.e. • and <0 • Define to be the that gives the inf value for . • Therefore, () - () • is a negative cycle, hence: • <0 • Therefore, • violates cyclic monotonicity .

  18. Computationally-Efficient Approximation Mechanisms • Cyclic monotonicity Corollary: A social choice function is dominant-strategy implementable it satisfies cyclic monotonicity Going back to our example: Two monotonicity conditions

  19. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Cyclic monotonicity Example: Should check: Set . The shortest path from to The shortest path from to 2 -1

  20. Computationally-Efficient Approximation Mechanisms • A social choice setting – reminder • Two monotonicity conditions • Cyclic monotonicity • representation graph of a social choice function • Weak monotonicity • Weak monotonicity in Order-based domain • Single-Dimensional Domains and Job Scheduling • Scheduling related machines • single-dimensional linear domains • Summary Outline:

  21. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Weak monotonicity Cyclic monotonicity: We found condition on involves only the properties of , without existential price qualifiers. Only: It is quite complex. k could be large, and a “shorter” condition would have been nicer. Definition:Weak monotonicity (W-MON) A social choice function satisfies W-MON if for every player , and , and with

  22. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Weak monotonicity Definition:Weak monotonicity (W-MON) A social choice function satisfies W-MON if for every player , and , and with If the outcome changes from to when changes her type from to , then ’s value for has increased at least as ’s value for in the transition to . Note: W-MON is a special case of Cyclic monotonicity when

  23. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Weak monotonicity W-MON is necessary for truthfulness. When is it also sufficient? Theorem: If the domain is convex, then any social choice function that satisfies W-MONis dominant-strategy implementable. We will prove it for special case: “base-order” domains. Fix player , some . W.l.o.g: : (otherwise we remove from for player )

  24. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Weak monotonicity Definition:Order-based domain A domain is “order-based” if there exists a partial orderover the set s.t : with . Example: = {chocolate, banana, apple} c a b

  25. Computationally-Efficient Approximation Mechanisms Two monotonicity conditions • Weak monotonicity Theorem: If the domain is ordered-based then any social choice function that satisfies W-MONis dominant-strategy implementable. Open problem: Exactly characterize the domains for which W-MON is sufficient for implementability.

  26. Computationally-Efficient Approximation Mechanisms • A social choice setting – reminder • Two monotonicity conditions • Cyclic monotonicity • representation graph of a social choice function • Weak monotonicity • Weak monotonicity in Order-based domain • Single-Dimensional Domains and Job Scheduling • Scheduling related machines • single-dimensional linear domains • Summary Outline:

  27. Computationally-Efficient Approximation Mechanisms Single-Dimensional Domains and Job Scheduling • Scheduling related machines n jobs m machines 00:00:01 00:31:08 00:00:07 1066 MHz 3060MHz 00:00:42 99:99:99 3000MHz 20:99:98 2Hz 99:99:99

  28. Computationally-Efficient Approximation Mechanisms Single-Dimensional Domains and Job Scheduling • Scheduling related machines jobs are to be assigned to machines, where job consumes time-units, and machine has speed . Thus machine requires time-units to complete job . Let be the load on machine . Goal: minimize (the makespan).

  29. Computationally-Efficient Approximation Mechanisms Single-Dimensional Domains and Job Scheduling • Scheduling related machines Each machine is selfish entity. Utility of a machine with a load and a payment : 1066 MHz 3060MHz 3000MHz 2Hz

  30. Computationally-Efficient Approximation Mechanisms Single-Dimensional Domains and Job Scheduling single-dimensional linear domains Definition:single-dimensional linear domains A domain of player is a single-dimensional linear domain if: (loads) s.t (cost) s.t Disclosing of player gives us the entire valuation vector. Machine scheduling is single-dimensional linear domain: For each ,, is the load of machine according to

  31. Computationally-Efficient Approximation Mechanisms Single-Dimensional Domains and Job Scheduling single-dimensional linear domains Goal: design a computationally-efficient approximation algorithm, that is also implementable. Can we use VCG? No: we have min-max and not minimize of sum of costs. We have convex domain we need a W-MON algorithm

  32. Computationally-Efficient Approximation Mechanisms Single-Dimensional Domains and Job Scheduling single-dimensional linear domains Remember: Definition:Weak monotonicity (W-MON) A social choice function satisfies W-MON if for every player , and , and with Assume W-MON:

  33. Computationally-Efficient Approximation Mechanisms Single-Dimensional Domains and Job Scheduling single-dimensional linear domains Remember: Theorem: If the domain is ordered-based then any social choice function that satisfies W-MONis dominant-strategy implementable. We got W-MON Such an algorithm is implementable its load functions are monotone non-increasing.

  34. Computationally-Efficient Approximation Mechanisms Single-Dimensional Domains and Job Scheduling single-dimensional linear domains From here one can show: Theorem: An algorithm for a single-dimensional linear domain is implementable load functions are non-increasing. Furthermore, if this is the case then charging from every player a price Finaly one can show A monotone algorithm for the job scheduling problem.

  35. Computationally-Efficient Approximation Mechanisms • A social choice setting – reminder • Two monotonicity conditions • Cyclic monotonicity • representation graph of a social choice function • Weak monotonicity • Weak monotonicity in Order-based domain • Single-Dimensional Domains and Job Scheduling • Scheduling related machines • single-dimensional linear domains • Summary Summary:

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